Coordination of multi-agent systems: stability via nonlinear Perron-Frobenius theory and consensus for desynchronization and dynamic estimation.

This thesis addresses a variety of problems that arise in the study of complex networks composed by multiple interacting agents, usually called multi-agent systems (MASs). Each agent is modeled as a dynamical system whose dynamics is fully described by a state-space representation. In the first part the focus is on the application to MASs of recent results that deal with the extensions of Perron-Frobenius theory to nonlinear maps. In the shift from the linear to the nonlinear framework, Perron-Frobenius theory considers maps being order-preserving instead of matrices being nonnegative. The main contribution is threefold. First of all, a convergence analysis of the iterative behavior of two novel classes of order-preserving nonlinear maps is carried out, thus establishing sufficient conditions which guarantee convergence toward a fixed point of the map: nonnegative row-stochastic matrices turns out to be a special case. Secondly, these results are applied to MASs, both in discrete and continuous-time: local properties of the agents' dynamics have been identified so that the global interconnected system falls into one of the above mentioned classes, thus guaranteeing its global stability. Lastly, a sufficient condition on the connectivity of the communication network is provided to restrict the set of equilibrium points of the system to the consensus points, thus ensuring the agents to achieve consensus. These results do not rely on standard tools (e.g., Lyapunov theory) and thus they constitute a novel approach to the analysis and control of multi-agent dynamical systems. In the second part the focus is on the design of dynamic estimation algorithms in large networks which enable to solve specific problems. The first problem consists in breaking synchronization in networks of diffusively coupled harmonic oscillators. The design of a local state feedback that achieves desynchronization in connected networks with arbitrary undirected interactions is provided. The proposed control law is obtained via a novel protocol for the distributed estimation of the Fiedler vector of the Laplacian matrix. The second problem consists in the estimation of the number of active agents in networks wherein agents are allowed to join or leave. The adopted strategy consists in the distributed and dynamic estimation of the maximum among numbers locally generated by the active agents and the subsequent inference of the number of the agents that took part in the experiment. Two protocols are proposed and characterized to solve the consensus problem on the time-varying max value. The third problem consists in the average state estimation of a large network of agents where only a few agents' states are accessible to a centralized observer. The proposed strategy projects the dynamics of the original system into a lower dimensional state space, which is useful when dealing with large-scale systems. Necessary and sufficient conditions for the existence of a linear and a sliding mode observers are derived, along with a characterization of their design and convergence properties

Dynamic Max-Consensus and Size Estimation of Anonymous Multi-Agent Networks

In this paper we propose a novel consensus protocol for discrete-time multi-agent systems (MAS), which solves the dynamic consensus problem on the max value, i.e., the dynamic max-consensus problem. In the dynamic max-consensus problem to each agent is fed a an exogenous reference signal, the objective of each agent is to estimate the instantaneous and time-varying value of the maximum among the signals fed to the network, by exploiting only local and anonymous interactions among the agents. The absolute and relative tracking error of the proposed distributed control protocol is theoretically characterized and is shown to be bounded and by tuning its parameters it is possible to trade-off convergence time for steady-state error. The dynamic Max-consensus algorithm is then applied to solve the distributed size estimation problem in a dynamic setting where the size of the network is time-varying during the execution of the estimation algorithm. Numerical simulations are provided to corroborate the theoretical analysis

Stability of Nonexpansive Monotone Systems and Application to Recurrent Neural Networks

This letter shows that trajectories of continuous-time monotone systems (in the sense of Kamke-Muller) converge to equilibrium points if their vector field is continuously differentiable and if they are nonexpansive w.r.t. a diagonally weighted infinity norm. Differently from the current literature trend, the system is not required to be contractive but merely nonexpansive, thus allowing for multiple equilibrium points. Easy-to-check conditions on the vector field to verify that the system is both monotone and nonexpansive are provided. This is done by showing that nonexpansiveness is implied by subhomogeneity of the system, a generalization of the translation invariance property. We apply the results in the context of RNNs, thus providing sufficient conditions for convergence of the state trajectories of nonexpansive monotone neural networks that are not contractive

Distributed Estimation of the Laplacian Spectrum via Wave Equation and Distributed Optimization

This paper presents a distributed algorithm to estimate all distinct eigenvalues of the Laplacian matrix encoding the unknown topology of a multi-agent system. The agents interact according to the discrete-time wave equation so that their state trajectory persistently oscillates with modes that depend on the eigenvalues of the Laplacian matrix. In this way, the problem of distributed estimation of the eigenvalues of the Laplacian is recast into that of estimating the modes of evolution of the state-trajectory of a linear dynamical system. Unlike previous literature, this paper formulates a distributed optimization problem where, by considering its own state trajectory, each agent estimates all distinct eigenvalues of the Laplacian matrix. The main advantages of the proposed algorithm are the ability of each agent to estimate also eigenvalues corresponding to modes unobservable from its own state trajectory, a much greater numerical stability, and therefore improved scalability to large networks wit h respect to competing approaches, as evidenced by the numerical comparisons

Online Distributed Learning over Random Networks

The recent deployment of multi-agent systems in a wide range of scenarios has enabled the solution of learning problems in a distributed fashion. In this context, agents are tasked with collecting local data and then cooperatively train a model, without directly sharing the data. While distributed learning offers the advantage of preserving agents' privacy, it also poses several challenges in terms of designing and analyzing suitable algorithms. This work focuses specifically on the following challenges motivated by practical implementation: (i) online learning, where the local data change over time; (ii) asynchronous agent computations; (iii) unreliable and limited communications; and (iv) inexact local computations. To tackle these challenges, we introduce the Distributed Operator Theoretical (DOT) version of the Alternating Direction Method of Multipliers (ADMM), which we call the DOT-ADMM Algorithm. We prove that it converges with a linear rate for a large class of convex learning problems (e.g., linear and logistic regression problems) toward a bounded neighborhood of the optimal time-varying solution, and characterize how the neighborhood depends on~$\text{(i)--(iv)}$. We corroborate the theoretical analysis with numerical simulations comparing the DOT-ADMM Algorithm with other state-of-the-art algorithms, showing that only the proposed algorithm exhibits robustness to (i)--(iv)

Scale-free estimation of the average state in large-scale systems

International audienceThis paper provides a computationally tractable necessary and sufficient condition for the existence of an average state observer for large-scale linear time-invariant (LTI) systems. Two design procedures, each with its own significance, are proposed. When the necessary and sufficient condition is not satisfied, a methodology is devised to obtain an optimal asymptotic estimate of the average state. In particular, the estimation problem is addressed by aggregating the unmeasured states of the original system and obtaining a projected system of reduced dimension. This approach reduces the complexity of the estimation task and yields an observer of dimension one. Moreover, it turns out that the dimension of the system also does not affect the upper bound on the estimation error

Azimuthal anisotropy of charged jet production in root s(NN)=2.76 TeV Pb-Pb collisions

We present measurements of the azimuthal dependence of charged jet production in central and semi-central root s(NN) = 2.76 TeV Pb-Pb collisions with respect to the second harmonic event plane, quantified as nu(ch)(2) (jet). Jet finding is performed employing the anti-k(T) algorithm with a resolution parameter R = 0.2 using charged tracks from the ALICE tracking system. The contribution of the azimuthal anisotropy of the underlying event is taken into account event-by-event. The remaining (statistical) region-to-region fluctuations are removed on an ensemble basis by unfolding the jet spectra for different event plane orientations independently. Significant non-zero nu(ch)(2) (jet) is observed in semi-central collisions (30-50% centrality) for 20 <p(T)(ch) (jet) <90 GeV/c. The azimuthal dependence of the charged jet production is similar to the dependence observed for jets comprising both charged and neutral fragments, and compatible with measurements of the nu(2) of single charged particles at high p(T). Good agreement between the data and predictions from JEWEL, an event generator simulating parton shower evolution in the presence of a dense QCD medium, is found in semi-central collisions. (C) 2015 CERN for the benefit of the ALICE Collaboration. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Peer reviewe

Production of He-4 and (4) in Pb-Pb collisions at root(NN)-N-S=2.76 TeV at the LHC

Results on the production of He-4 and (4) nuclei in Pb-Pb collisions at root(NN)-N-S = 2.76 TeV in the rapidity range vertical bar y vertical bar <1, using the ALICE detector, are presented in this paper. The rapidity densities corresponding to 0-10% central events are found to be dN/dy4(He) = (0.8 +/- 0.4 (stat) +/- 0.3 (syst)) x 10(-6) and dN/dy4 = (1.1 +/- 0.4 (stat) +/- 0.2 (syst)) x 10(-6), respectively. This is in agreement with the statistical thermal model expectation assuming the same chemical freeze-out temperature (T-chem = 156 MeV) as for light hadrons. The measured ratio of (4)/He-4 is 1.4 +/- 0.8 (stat) +/- 0.5 (syst). (C) 2018 Published by Elsevier B.V.Peer reviewe

Pseudorapidity and transverse-momentum distributions of charged particles in proton-proton collisions at root s=13 TeV

The pseudorapidity (eta) and transverse-momentum (p(T)) distributions of charged particles produced in proton-proton collisions are measured at the centre-of-mass energy root s = 13 TeV. The pseudorapidity distribution in vertical bar eta vertical bar <1.8 is reported for inelastic events and for events with at least one charged particle in vertical bar eta vertical bar <1. The pseudorapidity density of charged particles produced in the pseudorapidity region vertical bar eta vertical bar <0.5 is 5.31 +/- 0.18 and 6.46 +/- 0.19 for the two event classes, respectively. The transverse-momentum distribution of charged particles is measured in the range 0.15 <p(T) <20 GeV/c and vertical bar eta vertical bar <0.8 for events with at least one charged particle in vertical bar eta vertical bar <1. The evolution of the transverse momentum spectra of charged particles is also investigated as a function of event multiplicity. The results are compared with calculations from PYTHIA and EPOS Monte Carlo generators. (C) 2015 CERN for the benefit of the ALICE Collaboration. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Peer reviewe